Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
A Robust Implementation for Solving the S-Unit Equation and Several Applications Read chapter Read first chapter

2021 | OriginalPaper | Chapter

Computing Rational Points on Rank 0 Genus 3 Hyperelliptic Curves

Authors : María Inés de Frutos-Fernández, Sachi Hashimoto

Published in: Arithmetic Geometry, Number Theory, and Computation

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We compute rational points on genus 3 odd degree hyperelliptic curves C over \(\mathbb {Q}\) that have Jacobians of Mordell–Weil rank 0. The computation applies the Chabauty–Coleman method to find the zero set of a certain system of p-adic integrals, which is known to be finite and include the set of rational points \(C(\mathbb {Q})\). We implemented an algorithm in Sage to carry out the Chabauty–Coleman method on a database of 5870 curves.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB
next chapter Curves with Sharp Chabauty-Coleman Bound
Footnotes
1
The global height is the absolute logarithmic height of the point, which is the maximum of the absolute logarithmic heights of its coordinates. For a rational point \(\frac {n}{d}\), this height is computed as \(\max (\log (|n|),\log (|d|)).\)
 
Literature
1.
2.
J. S. Balakrishnan, F. Bianchi, V. Cantoral-Farfán, M. Çiperiani, and A. Etropolski, “Chabauty-Coleman experiments for genus 3 hyperelliptic curves,” in Research Directions in Number Theory, ser. Association for Women in Mathematics Series. Springer, 2019, vol. 19, pp. 67–90.CrossRef
3.
J. S. Balakrishnan, R. W. Bradshaw, and K. S. Kedlaya, “Explicit Coleman integration for hyperelliptic curves,” in Algorithmic Number Theory. ANTS 2010, Lecture Notes in Computer Science. Springer, Berlin, 2010, vol. 6197, pp. 16–31.
4.
A. R. Booker, J. Sijsling, A. V. Sutherland, J. Voight, and D. Yasaki, “A database of genus-2 curves over the rational numbers,” LMS J. Comput. Math., vol. 19, no. suppl. A, pp. 235–254, 2016.
5.
W. Bosma, J. Cannon, and C. Playoust, “The Magma Algebra System. I. The User Language,” J. Symbolic Comput., vol. 24, no. 3-4, pp. 235–265, 1997.MathSciNetCrossRef
6.
C. Chabauty, “Sur les points rationnels des courbes algébriques de genre supérieur à l’unité,” C. R. Acad. Sci. Paris, vol. 212, pp. 882–885, 1941.MathSciNetMATH
7.
8.
——, “Torsion points on curves and p-adic abelian integrals,” Ann. of Math. (2), vol. 121, no. 1, pp. 111–168, 1985.MathSciNetCrossRef
9.
10.
G. Faltings, “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern,” Invent. Math., vol. 73, no. 3, pp. 349–366, 1983.MathSciNetCrossRef
11.
W. McCallum and B. Poonen, “The method of Chabauty and Coleman,” Explicit Methods in Number Theory, Rational Points and Diophantine Equations, Panor. Synthèses, vol. 36, pp. 99–117, 2012.MathSciNetMATH
12.
J. S. Milne, “Jacobian varieties,” in Arithmetic geometry (Storrs, Conn., 1984). Springer, New York, 1986, pp. 167–212.
13.
M. Stoll, “On the height constant for curves of genus two. II,” Acta Arith., vol. 104, no. 2, pp. 165–182, 2002.MathSciNetCrossRef
14.
——, “Independence of rational points on twists of a given curve,” Compos. Math., vol. 142, no. 5, pp. 1201–1214, 2006.MathSciNetCrossRef
15.
16.
17.
J. L. Wetherell, Bounding the Number of Rational Points on Certain Curves of High Rank, 1997, Ph.D. thesis, University of California at Berkeley.
Metadata
Title
Computing Rational Points on Rank 0 Genus 3 Hyperelliptic Curves
Authors
María Inés de Frutos-Fernández
Sachi Hashimoto
Copyright Year
2021
Book
Arithmetic Geometry, Number Theory, and Computation

Print ISBN: 978-3-030-80913-3

Electronic ISBN: 978-3-030-80914-0

Copyright Year: 2021

DOI
https://doi.org/10.1007/978-3-030-80914-0_14

Premium Partner

    Image Credits